Linear optimization: Parametrical objective function 4: Unterschied zwischen den Versionen
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Compared to the [[sensitivity analysis]] the Parametrical objective function ''makes a statement about large changes in the input data''. | Compared to the [[sensitivity analysis]] the Parametrical objective function ''makes a statement about large changes in the input data''. | ||
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− | To include large changes in the input data you have to add a new variable "q". For simple cases you just summate the new variable "q" multiplicated with a constant vektor " | + | == Basic Knowledge == |
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+ | To include large changes in the input data you have to add a new variable "<math>q</math>". For simple cases you just summate the new variable "<math>q</math>" multiplicated with a constant vektor " <math>\beta</math> " to the objective function. In this case the constant vektor is the value which change the input parameter "<math>c</math>". | ||
The objectiv function is now with a parameter: | The objectiv function is now with a parameter: | ||
− | c(q)=c+q* | + | <math>c(q)=c+q*</math><math>\beta</math> |
Thereby there is a new optimization problem which can be solved. | Thereby there is a new optimization problem which can be solved. | ||
− | + | <math>P(q)</math><math>=</math><math>(c+</math><math>q</math><math>\beta)</math><math>^T x</math> <math>\rightarrow</math> <math>max!</math> | |
<math>Ax=b</math> | <math>Ax=b</math> | ||
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<math>x</math> <math>\ge</math> <math>0</math> | <math>x</math> <math>\ge</math> <math>0</math> | ||
− | =Exemplification= | + | == Exemplification == |
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+ | In the follow we are going to show an example of "Linear Optimization of Parametrical Objective Function". | ||
+ | We want to find an optimal combination of product 1 and product 2 to minimize the costs. | ||
+ | Diffrent from "normal" simplex are the parameters in the objective function. | ||
+ | This parameters express a modification in the objective funktion, it is usefull for changing the input data without calculating the whole simplex again. | ||
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+ | [[Datei:Unbenannt.PNG]] | ||
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+ | From this tableau you can read out the profit function with a variable inside: | ||
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+ | <math>G(q)= q*x_1+500x_2-36000</math> | ||
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+ | Transformed into the Simplex-Tableau: | ||
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+ | [[Datei:Tabelle_1.PNG]] (Tableau 1) | ||
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+ | In Tableau 1 and 2 you do the normal simplex iteration. "Ignoring" the variable "<math>q</math>", setting it equal "0". | ||
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+ | [[Datei:Tabelle_2.PNG]] (Tableau 2) | ||
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+ | [[Datei:Tabbelle_3.PNG]] (Tableau 3) | ||
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+ | In Tableau 3 you have an optimal solution if you define <math>q</math> <math>\le</math> <math>0</math> ,because your head row gets positive <math>\rightarrow</math> acceptable and optimal solution | ||
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+ | == Origin == | ||
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+ | -Skript OperationResearch_2012SS_Wendt | ||
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+ | - http://www.orklaert.de/parametrische-lineare-optimierung | ||
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+ | - https://bisor.wiwi.uni-kl.de/orwiki/Parametrische_Optimierung | ||
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+ | ==Made By== | ||
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+ | Matrknr.: 381935; 380535 |
Aktuelle Version vom 28. Juli 2013, 21:43 Uhr
Parametrical objective function is a special part of linear optimization. The foundation of parametrical optimization is the sensitivity analysis. Compared to the sensitivity analysis the Parametrical objective function makes a statement about large changes in the input data.
Inhaltsverzeichnis
Basic Knowledge
To include large changes in the input data you have to add a new variable "". For simple cases you just summate the new variable "" multiplicated with a constant vektor " " to the objective function. In this case the constant vektor is the value which change the input parameter "".
The objectiv function is now with a parameter:
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): c(q)=c+q*
Thereby there is a new optimization problem which can be solved.
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \rightarrow
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \ge
Exemplification
In the follow we are going to show an example of "Linear Optimization of Parametrical Objective Function". We want to find an optimal combination of product 1 and product 2 to minimize the costs. Diffrent from "normal" simplex are the parameters in the objective function. This parameters express a modification in the objective funktion, it is usefull for changing the input data without calculating the whole simplex again.
From this tableau you can read out the profit function with a variable inside:
Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): G(q)= q*x_1+500x_2-36000
Transformed into the Simplex-Tableau:
In Tableau 1 and 2 you do the normal simplex iteration. "Ignoring" the variable "", setting it equal "0".
In Tableau 3 you have an optimal solution if you define Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \le
,because your head row gets positive Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \rightarrow
acceptable and optimal solution
Origin
-Skript OperationResearch_2012SS_Wendt
- http://www.orklaert.de/parametrische-lineare-optimierung
- https://bisor.wiwi.uni-kl.de/orwiki/Parametrische_Optimierung
Made By
Matrknr.: 381935; 380535